用 (use) is the indexing Kangxi radical and supplies the second half of the meaning. The character is a textbook 会意字: stack 不 over 用 and read it bù-yòng → béng. Spoken pronunciation came first; the written character was reverse-engineered to match.
不 sits on top supplying the negation. 甭 is a transparent compound ideograph: 不 (not) plus 用 (need) literally fused into béng, the Beijing-dialect contraction of bù yòng (don't need to / no need). The pronunciation itself is a phonetic squish of 不用.
